Central factor satisfies image condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., image condition)
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Statement

Statement with symbols

Suppose $H$ is a central factor of a group $G$ (in other words, $H C_{G} (H) = G$ ). Suppose $φ : G \to K$ is a surjective homomorphism of groups. Then, $φ (H)$ is a central factor of $K$ .

Definitions used

Central factor

Further information: Central factor

A subgroup $H$ of a group $G$ is termed a central factor of $G < / m a h > i f i t s a t i s f i e s t h e f o l l o w i n g e q u i v a l e n t c o n d i t i o n s : * < m a t h > H C_{G} (H) = G$ , where $C_{G} (H)$ denotes the centralizer of $H$ in $G$ .

Every inner automorphism of $G$ restricts to an inner automorphism of $H$ .

Related facts

Similar facts about related properties

Proof

Proof in terms of centralizers

Given: A central factor $H$ of a group $G$ . A surjective homomorphism $φ : G \to K$ .

To prove: $φ (H) C_{K} (φ (H)) = K$ .

Proof: By the definition of homomorphism, if two elements commute in $G$ , their images commute in $K$ . Thus, the definition of centralizer yields:

$φ (C_{G} (H)) \leq C_{K} (φ (H))$ .

Taking the product of both sides with $φ (H)$ yields:

$φ (H) φ (C_{G} (H)) \subseteq φ (H) C_{K} (φ (H))$ .

By the definition of homomorphism, the left side is the same as $φ (H C_{G} (H))$ , which is $φ (G)$ (since $H$ is a central factor of $G$ ). $φ (G) = K$ by the assumption of surjectivity, so we get:

$K \subseteq φ (H) C_{K} (φ (H)) \subseteq K$ .

This forces:

$φ (H) C_{K} (φ (H)) = K$

completing the proof.